Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device
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Rock Mechanics and Rock Engineering manuscript No.
(will be inserted by the editor)
Inducing tensile failure of claystone through thermal pressurization in a
novel triaxial device
Philipp Braun · Pierre Delage · Siavash Ghabezloo ·
Baptiste Chabot · Nathalie Conil · Minh-Ngoc Vu
arXiv:2203.09828v1 [physics.geo-ph] 18 Mar 2022
Accepted manuscript DOI: 10.1007/s00603-022-02838-3
Highlights
– Thermal pressurization of clay rock in deep radioactive waste repositories can reduce the effective stresses,
which can lead to damage or failure.
– Our novel laboratory triaxial device is able mimic in situ conditions: Constant vertical total stress, zero
lateral deformation and thermal pressurization.
– Pore pressure increase, vertical extension strains and thermal pressurization failure were recorded in a series
of tests on Callovo-Oxfordian claystone specimens.
– The effective tensile strength was reached at values around 3 MPa in tension and temperatures between 53
and 64 °C, creating sub-horizontal fractures.
– The experimental responses can be well reproduced using a thermo-poroelasticity model.
– Hoek-Brown and Fairhurst generalized Griffith criteria appear suitable to account for the rock’s tensile
resistance.
P. Braun · S. Ghabezloo · P. Delage · B. Chabot ·
Laboratoire Navier, 6-8 avenue Blaise-Pascal, Cité Descartes 77455 Champs-sur-Marne, Paris, France
E-mail: philipp.braun@enpc.fr
P. Braun · M.-N. Vu
Andra, 1 Rue Jean Monnet, 92290 Châtenay-Malabry, France
N. Conil
Andra, Centre de Meuse/Haute-Marne BP9, 55290 Bure, France
Now at: INERIS, Ecole des Mines, Parc de Saurupt, 54042 Nancy, France2 Philipp Braun et al. Abstract Complex coupled thermo-hydromechanical (THM) loading paths are expected to occur in clay rocks which serve as host formations for geological radioactive waste repositories. Exothermic waste packages heat the rock, causing thermal strains and temperature induced pore pressure build-up. The drifts are designed in such a way as to limit these effects. One has to anticipate failure and fracturing of the material, should pore pressures exceed the tensile resistance of the rock. To characterise the behaviour of the Callovo-Oxfordian claystone (COx) under effective tension and to quantify the tensile failure criterion, a laboratory program is carried out in this work. THM loading paths which correspond to the expected in situ conditions are recreated in the laboratory. To this end, a special triaxial system was developed, which allows the independent control of radial and axial stresses, as well as of pore pressure and temperature of rock specimens. More importantly, the device allows one to maintain axial effective tension on a specimen. Saturated cylindrical claystone specimens were tested in undrained conditions under constrained lateral deformation and under nearly constant axial stress. The specimens were heated until the induced pore pressures created effective tensile stresses and ultimately fractured the material. The failure happened at average axial effective tensile stresses around 3.0 MPa. Fracturing under different lateral total stresses allows one to describe the failure with a Hoek-Brown or Fairhurst’s generalized Griffith criterion. Measured axial extension strains are analysed based on a transversely isotropic thermo- poroelastic constitutive model, which is able to satisfactorily reproduce the observed behaviour. Keywords Thermal pressurization · Tensile failure · Thermo-poroelasticity · Transverse isotropy · Callovo- Oxfordian claystone · Nuclear waste disposal
Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 3 List of Symbols Note that the matrix notation is used throughout this work. h Direction parallel to bedding z Direction perpendicular to bedding σi Total stress in direction i σi0 Terzaghi effective stress in direction i εi Strain in direction i σc Unconfined compressive strength σt Tensile strength Ei Young’s modulus in direction i νi Poisson’s ratio in direction i mi Hoek-Brown criterion parameter φ Porosity ρ Wet density ρd Dry density w Water content Sr Degree of saturation s Suction pf Pore pressure T Temperature σ Isotropic confining pressure εhyd Volumetric hydration swelling γt Fracture angle with respect to bedding Cij Elastic compliance matrix bi Biot’s coeffcient in direction i αd,i Drained thermal expansion coeffcient in direction i G0 Shear modulus within the isotropic plane G Shear modulus perpendicular to the isotropic plane αφ Bulk thermal expansion coefficient of the pore volume M Biot’s modulus Kf Bulk modulus of the pore fluid Kφ Bulk modulus of the pore volume ϕ Friction angle c Cohesion VL Volume of the drainage system cL Compressibility of the drainage system with respect to fluid pressure pL Fluid pressure within the drainage system κL Compressibility of the drainage system with respect to radial confining pressure σrad Radial confining pressure αL Bulk thermal expansion coefficient of the drainage system Ms Fluid mass within the specimen mf Fluid mass per unit volume of the specimen Vs Total specimen volume ML Fluid mass within the drainage system ρL Fluid density within the drainage system ρf Pore fluid density cf Pore fluid compressibility αf Pore fluid bulk thermal expansion coefficient
4 Philipp Braun et al.
1 Introduction
A detailed investigation of the thermo-poroelastic properties of host rocks is essential in the design of the
deep geological disposals of high-level radioactive wastes in clay rocks, as considered in France, Switzerland
and Belgium. Next to the hydro-mechanical response involved during the excavation of the micro-tunnels, a
thermo-hydro-mechanical response takes place during to the heat release from the exothermic waste packages
(e.g. Gens et al., 2007; Seyedi et al., 2017; Armand et al., 2017; Conil et al., 2020, see Fig. 1). Heat radiates from
the packages, resulting in gradually increasing temperature in the rock. This so-called thermal pressurization
occurs when saturated soils or rocks are heated in undrained conditions, due to the significant difference between
the thermal expansion coefficients of water and of the solid phase (Ghabezloo and Sulem, 2009; Vu et al., 2020).
Quasi-undrained conditions are provided through the very low permeability of the claystone (in the order of
10−20 m2 , Escoffier et al., 2005; Davy et al., 2007; Menaceur et al., 2015, 2016).
Thermal pressurization has been observed in various in situ heating tests carried out in underground re-
search laboratories (URL) excavated in claystones. It has been described and numerically modelled by dif-
ferent authors (e.g. by Gens et al., 2007; Jobmann and Polster, 2007 in the HE-D test run in the Opalinus
clay in the Mt Terri URL in Switzerland, by Seyedi et al., 2017; Conil et al., 2020 in the TED test and
by Armand et al. (2017); Bumbieler et al. (2021); Tourchi et al. (2021) in the ALC1604 test, both run in the
Callovo-Oxfordian claystone (COx) in the Bure URL in France). Thermal pressurization has also been inves-
tigated through laboratory experiments, as done by Mohajerani et al., 2012; Zhang et al., 2017; Braun et al.,
2021a on COx claystone and Monfared et al., 2011b on Opalinus clay.
The thermo-poromechanical strain response of claystones in drained and undrained conditions has been
investigated in the laboratory by various authors including Zhang et al., 2017; Mohajerani et al., 2014; Menaceur
et al., 2016; Belmokhtar et al., 2017b,a; Braun et al., 2020, 2021a on the COx claystone and Monfared et al.,
2011b and Favero et al., 2016 on the Opalinus clay. A thermo-elastoplastic behaviour upon drained heating
under constant effective stress has been evidenced by Monfared et al., 2011b on the Opalinus clay and by
Belmokhtar et al., 2017a; Braun et al., 2020, 2021a on the COx claystone. The latter authors found that both
elastic and plastic drained strains were anisotropic during the tests. These features are somewhat similar to
what has been observed in previous investigations on clays (e.g. Baldi et al., 1988; Sultan et al., 2002 on Boom
clay and Abuel-Naga et al., 2007 on Bangkok clay).
Figure 1a illustrates schematically the current French concept for high level radioactive waste disposal at
great depth, in which waste canisters are deposited in parallel horizontal sleeved micro-tunnels of 0.7 to 1.0
m diameter and 80 to 150 m length (Armand et al., 2015; Plúa et al., 2021a,b). The micro-tunnels are evenly
spaced with enough distance such as to keep the temperature below 90 °C in the rock and to ensure no damage
in the far field. Given the length of the micro-tunnels and their periodic layout, thermo-hydro-mechanical
processes are often modelled assuming plane strain conditions (Fig. 1b), with symmetry conditions at mid-
distance between them. This results in peculiar conditions at this location, i.e. zero horizontal deformations
and no horizontal heat nor fluid flux. One can expect a significant thermal pore pressure build-up in this area,
since the horizontal fluid flow is limited through the no-flux boundary conditions. With time, thermal pore
pressures dissipate towards the far-field. In summary, the rock mass at this mid-distance is submitted to zero
lateral deformation, constant overburden vertical total stress and temperature increase. Constrained lateral
thermal expansion leads to an increase of lateral total stress. In such conditions, one has to consider the risk
of reaching a pore pressure level larger than the total vertical stress, which could lead to local vertical tensile
failure (Li and Wong, 2018). Note that the planned design of the high level radioactive waste repository does
not allow any tensile failure. More importantly, the study of thermally induced fracturing is crucial for risk
management.
The tensile strength of clay-rocks has been widely investigated in rock mechanics, and commonly measured
through the direct tension test (Hoek, 1964; Brace, 1964) or the Brazilian test (Akazawa, 1943; Carneiro, 1943;
ISRM, 1978). Yang and Hsieh (1997) determined the tensile strength of a transversely isotropic claystone in
direct tension tests. They evidenced a smaller tensile resistance for tension applied perpendicular to the planeInducing tensile failure of claystone through thermal pressurization in a novel triaxial device 5
a)
Waste packages
Access gallery
Micro-tunnels
Cross-section
Temperature
b)
Natural Heated
Symmetry-axis:
-Zero horizontal deformation
-Zero horizontal heat or fluid flow
Fig. 1 a) Layout of the micro-tunnels containing high-level radioactive waste (adopted from Andra, 2005), with b) a cross-
section where one can assume plane strain and zero heat and temperature flux perpendicular to the image plane, illustrating
the heat radiation from periodically constructed micro-tunnels.
of isotropy (∼ 4 MPa) with respect to that measured on specimens loaded parallel to the plane of isotropy
(∼ 11.5 MPa). Interestingly, under tension loading, they found elastic material properties close to the values
measured under compression. Only the Young modulus perpendicular to the plane of isotropy was detected to
be higher in tension than in compression. They observed a quite linear stress-strain behaviour under tension.
Coviello et al. (2005) carried out various different types of tensile tests, to determine the strength of two types
of weak rocks, Gasbeton and calcarenite. In the case of calcarenite, they found the same tensile strength with
both direct tensile test and Brazilian test (∼ 0.6 MPa). In the case of Gasbeton, the strength measured with
direct tension (∼ 0.9 MPa) was higher than measured in Brazilian tests (∼ 0.5 MPa). They noted that for both
materials, the Young moduli in tension and compression were about the same, with a very linear stress-strain
behaviour until failure. In the work of Hansen and Vogt (1987), an overview of various shales and claystones is
given with their unconfined compressive strength, elastic properties and tensile strength, presented in Tab. 1.
These are compared with the data from Bossart (2011) on the Opalinus clay.
According to Diederichs (2007), one can estimate σt = σc /mi , where σt is the tensile strength, σc the
unconfined compressive strength and mi a material constant of the Hoek-Brown model. An estimation for
claystone of mi ≈ 4.0±2 is recommended by Hoek (2007). Even though Perras and Diederichs (2014) concluded
that this formula does not give accurate results, it can be used to get a first approximation of the tensile
resistance of brittle rocks. The calculated values for mi of some shales and claystones are presented in Tab. 1,
confirming that they are generally higher than the recommended value of 4.0 ± 2. One observes a rather large6 Philipp Braun et al.
Table 1 Shale and claystone characteristics adopted from the literature, with an estimation of the Hoek-Brown coefficient
mi = σc /σt , compared with the experimental results of this study on COx claystone
Rock type σc E ν σt mi
[MPa] [GPa] [-] [MPa] [-]
Pierre shale1 7.2 0.6 0.12 0.5 13.6
Rhinestreet shale1 58.7 17.5 0.19 8.4 7.0
Green river shale1 94.8 10.4 0.31 11.9 8.0
Carlile shale1 22.8 3.3 0.16 3.5 6.5
Colorado shale2 2.97 2.4 0.42 0.4 6.9
Opalinus clay ⊥ 3 25.6 2.8 0.33 2.0 12.8
Opalinus clay k 3 10.5 7.2 0.24 1.0 10.5
COx claystone k 17.84 5.75 0.295 3.06 5.156
1 Hansen and Vogt (1987),
2 Mohamadi and Wan (2016), 3 Bossart (2011),
4 Andra database, 5 Braun et al. (2021b), 6 this study
variability of mi for these clay rocks between 6.5 and 13.6. Among the aforementioned studies, authors did not
investigate a temperature dependency of the tensile resistance.
The possibility of fracturing claystones through temperature-induced pore pressures has been recently
demonstrated by Li and Wong (2018) through laboratory unconfined heating tests. In lack of any stress and
strain measurements, these authors supposed that the fractures, that propagated parallel to the bedding plane,
were induced when the pore pressure exceeded the tensile strength. To the authors best knowledge, the thermal
fracturing of geomaterials has hitherto not been investigated under laboratory conditions of controlled stresses,
pore pressure and temperature. This paper presents the development of a novel thermal extension triaxial
device aimed at mimicking the thermal extension failure phenomenon under particular stress paths. The test
results show the THM response of the COx claystone from the intact to the fractured state when subjecting
it to thermal pressurization and provide an appropriate criterion accounting for its thermally induced tensile
failure.
2 Novel experimental device
This work’s aim was to develop a thermal extension triaxial device, in which a specimen should be submitted in
undrained conditions to a temperature increase. At the same time, radial strain has to remain zero and vertical
total stress constant, which corresponds to the in situ overburden stress. The zero radial stress condition involves
a servo-control of the confining pressure, while maintaining pore pressures larger than the total vertical stress
is another experimental challenge.
Similar to previous thermal testing devices developed and used in the same research group at Ecole des
Ponts ParisTech (e.g. Menaceur et al., 2016; Belmokhtar et al., 2017a; Braun et al., 2021a, a triaxial cell (Fig.
2a) was employed for the laboratory tests. A pressure volume controller (PVC1, GDS brand) was used to apply
pressure to the axial piston and to control the axial stress. Additional PVCs (PVC2,3, GDS brand) were used to
control the confining pressure and the pore pressure separately. Temperature changes were applied by using an
electric silicone heating belt, wrapped around the steel cell, and measured by means of a thermocouple located
inside the cell. Local (radial and axial) strains were accurately monitored by means of strain gages glued directly
onto the sample (see Braun et al., 2019). Accuracy in radial strain measurement is here of utmost importance
so as to ensure satisfactory zero lateral strain during heating.Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 7 Fig. 2 a) Schematic view of the triaxial cell. b) Zoom on the specimen assembly with novel extension modifications, permitting to maintain pore pressures larger than the axial total stress through glued connections, while saturation and drainage is achieved through lateral geotextiles. c) Photo showing a dummy specimen mounted with bottom and top caps and the axial piston. The aluminium force transducer is not shown in the photo. 2.1 Zero radial strain condition The zero radial strain condition was achieved through a servo-controlling routine programmed in the LabView software. The routine couples the strain measurements and the cell pressure PVC. Since the cell temperature changes during a test, it is required to correct the measured strains in real-time for thermally induced measure- ment errors. Following a correction method presented by Braun et al. (2019), a reference gage on a piece of steel 316L was placed in the triaxial cell next to the tested specimen. Doing so, thermal and mechanical strains on a reference material can be recorded. By knowing the thermal expansion coefficient and compressibility of steel 316L, the errors induced by any changes in cell temperature, room temperature, and confining pressure, can be determined as the difference between the measured reference strain and the calculated steel strain. This error was then subtracted from the strain measurements on the specimen. Braun et al. (2019) checked the quality of the thermal strain measurements by testing a specimen of aluminium alloy 2011 under thermal loads. A linear thermal expansion coefficient of the aluminium specimen of 2.30 × 10−5 ◦ C−1 was evaluated after correction, which fairly well corresponded with that provided by the manufacturer.
8 Philipp Braun et al.
During the laterally constrained heating tests, the corrected lateral strains were monitored, which were
maintained at zero value through a partial-integral-derivative (PID) software controller. This servo-control
routine calculated the required lateral stress increment, which was then transmitted to the PVC to change the
confining stress accordingly. The pressure applied by the axial PVC was simultaneously controlled in order to
maintain constant the axial stress. Tuning of the PID controller parameters was carried out before the tests.
It was possible to adapt them during the tests, so as to ensure a fast response of the confining PVC and to
avoid overshoot and oscillations. The PID routine was executed in intervals of 30 seconds, which proved being
sufficiently fast with respect to the adopted heating rate of maximum 10 ◦ C/h. The derivative part of the PID
was not necessary to achieve precise control.
2.2 Application of tensile axial stress
The main issue met in tension tests in triaxial cells for soils and rocks is the loss of contact between the axial
piston and the top cap, once the axial stress becomes smaller than the radial one (i.e. the confining pressure).
Donaghe and Chaney (1988) presented a review of the different available systems to avoid this loss of contact
in the case of soil testing. They mention screw connections, resin pot systems or a suction cap. The former
two methods are difficult to implement when the piston is not easily accessible once the contact with the top
cap is made. The suction cap, however, allows closing the cell and moving the piston downwards. Once in
contact, a certain surface between the top cap and piston is isolated from the confining stress by means of a
rubber membrane. By ensuring a lower pressure than the cell pressure within this surface, and given that the
isolated surface is larger than the specimen cross section, the confining pressure keeps the two pieces compressed
together. This allows one to transmit tension to the specimen. In the presented system, the connection between
the piston and the top cap was based on the principle of a suction cap, by using a flat disc shaped volume
isolated by means of an O-ring and connected to atmospheric pressure (Fig. 2b,c). As required, the isolated
volume has a base surface larger than the specimen cross section. As seen in Fig. 2b,c, a protrusion in the
piston ensures alignment of cap and piston.
Another issue is the loss of contact between top cap and specimen, once the pore pressure becomes equal
to the total axial stress. Pore pressures higher than the axial stress cannot be maintained in a device without
adequate modifications. A comprehensive overview of different methods to overcome this problem has been
given by Perras and Diederichs (2014). They recalled that the two main techniques for direct tensile testing
consisted either in attaching the specimen ends to the load frame (by using grips or by gluing, cf. Fairhurst,
1961; Hawkes and Mellor, 1970), or in modifying the sample shape to a so-called dog-bone shape (Hoek, 1964;
Brace, 1964). The latter method is not recommended for weak rocks such as claystones, as these materials
would not withstand the required shaping process in lathe (Perras and Diederichs, 2014). The former method
of gluing or gripping specimen has the advantage that conventional cylindrical specimens can be used. Hawkes
and Mellor (1970) have shown that strain inhomogeneities could be reduced, when the specimen was glued only
on its end surfaces and not along lateral sections. It was still likely that failure occurred close to the glued end
surfaces. In the presented device, the specimen and the caps were therefore glued together by tensile-resistant
epoxy resin.
Moreover, an internal axial force transducer was integrated between the specimen and the top cap. Given
that no convenient 20 mm diameter force gage was commercially available, a cylindrical aluminium (alloy
2011) piece of 30 mm height and 20 mm diameter was equipped with an axial strain gage (Fig. 2b). This force
transducer was previously calibrated, taking into account its thermal dilation.
In the final assembly (Fig. 2b), the aluminium cylinder was glued on top of the specimen. Afterwards,
two specially designed screw-able disk-shaped steel platens were glued onto the top of the force transducer
and onto the bottom end-surface of the specimen. A two-component epoxy resin (Araldite 2014, Huntsman
brand) was used, with a tensile strength of 26 MPa, much larger than the expected strength of the tested
specimens. Its glass transition temperature, up to which the resin maintains its original strength, is 80°C. ThisInducing tensile failure of claystone through thermal pressurization in a novel triaxial device 9
temperature that was not exceeded during the tests. Once the resin hardened, the assembly was screwed to
the bottom cap, which was fixed to the triaxial cell. The top cap was also screwed onto the assembly, allowing
us to maintain tensile stresses. The drainage lines drilled into the bottom platten were connected via the top
and bottom cap to a PVC that controls the pore pressure. Since the top and bottom surfaces of the specimen
have been made impermeable by impregnating with epoxy resin, drainage had to be ensured by means of a
geotextile wrapped around the specimen. The geotextile was connected to the lateral drainage line of the top
and bottom platens. Monfared et al. (2011b) have verified that these geotextiles remain permeable even under
high confining pressure. A neoprene membrane was then put over the specimen and the geotextile, isolating
the confining fluid (silicone oil) from the pore fluid.
The system was developed for specimens with a reduced diameter of 20 mm so as to enable faster drainage
and saturation (see Belmokhtar et al., 2018). The drainage length corresponds to the specimen radius of 10
mm and the height of the specimens was close to 35 mm.
2.3 Calibration of the drainage system
Ghabezloo and Sulem (2010) summarized the different properties of the drainage system, which are the volume
of the drainage system VL , its compressibility cL due to pressure changes in the drainage system pL , the
compressibility κL due to radial confining pressure changes dσrad , and the bulk thermal expansion coefficient
αL under a temperature change dT . The change in volume of the drainage system can hence be expressed as:
dVL
= cL dpL + αL dT − κL d σ rad (1)
VL
with the different parameters defined as:
1 ∂ VL
cL = (2)
VL ∂ pL T, σ rad
1 ∂ VL
αL = (3)
VL ∂T pL , σ rad
1 ∂ VL
κL = − (4)
VL ∂ σ rad T, pL
Pseudo-undrained conditions are here defined as a constant sum of fluid mass dMf = dMs + dML = 0
(where Ms = mf Vs is the fluid mass in the specimen, mf the specimen fluid mass per unit volume, Vs the
specimen total initial volume, ML = VL ρL the fluid mass in the drainage system and ρL the density of the
fluid in the drainage system). Rewriting the equations of Ghabezloo and Sulem (2010), one obtains:
VL ρf [(cL + cf ) dpL + (αL − αf ) dT − κL dσrad ] + Vs dmf = 0 (5)
where ρf , cf and αf are the porefluid density, compressibility and bulk thermal expansion coefficient, re-
spectively. In the presented experimental configuration drainage occurs through the lateral specimen surface,
therefore the fluid pressure in the drainage system pL is equal to the specimen pore pressure on this surface.
Calibration tests were carried out to characterize the properties of the drainage system, assuming that
they are constant with stress, pore pressure and temperature. To this end, a steel dummy specimen with zero
porosity (dMs = 0) was installed. Saturating the empty drainage system with water provided VL = 3500 mm3 .
Pore pressure changes were applied, while measuring dVL with PVC3, which allowed us to calculate cL = 1.33
GPa−1 . Changing the radial stress and measuring the pore pressure change in a closed drainage system resulted
in κL = 0.44 GPa−1 . The parameter αL = 0.79×10−4 °C−1 was determined by recording pore pressure changes
during temperature cycles.10 Philipp Braun et al.
3 Specimen characterization and preparation
The specimens of the COx claystone come from horizontal cores (EST 53650 and EST 57185) extracted at a
depth of 490 m in the Bure URL. The clay content of the COx claystone at this level is around 42 %, with an
average porosity of 17.5 % and an average water content of 7.9 % (Robinet et al., 2012; Conil et al., 2018). The
in situ stress conditions were measured by Wileveau et al. (2007), with a vertical and a minor horizontal total
stress of around 12 MPa, a major horizontal stress of around 16 MPa and a hydrostatic pore pressure of about
4.9 MPa.
Avoiding the desaturation and mechanical damage of cores is an important concern, since the claystone
mechanical properties should be kept as close as possible to the natural ones. Shales and claystones are known
to be particularly sensitive to changes in water content (Chiarelli et al., 2003; Valès et al., 2004; Zhang and
Rothfuchs, 2004; Pham et al., 2007; Zhang et al., 2012), that may occur during the successive processes of coring,
storage, transportation and trimming of laboratory specimens (Chiu et al., 1983; Monfared et al., 2011a; Ewy,
2015; Wild et al., 2017). Great attention has hence to be paid to preserve the initial water content of the cored
specimens. To avoid any contact with water, Andra is used to carry out coring with air-cooling, prior to sealing
the extracted cores in so-called T1 cells (Conil et al., 2018). T1 cells accommodate a 320 mm long COx core
with 80 mm diameter, previously wrapped in aluminium foil and in a latex membrane to avoid drying. A larger
diameter PVC tube is placed around the core and cement is cast in the annulus between the membrane and
the tube, creating a rigid mechanical protection. After cement hardening, a metal spring is used to constrain
the core along the axial direction.
Once arrived in the laboratory, T1 cells were opened and the cores were immediately covered by a layer
of paraffin wax, so as to prevent drying during the trimming process. Cylindrical proofs of 20 mm diameter
were drilled perpendicular to the bedding plane by means of a diamond coring bit cooled by compressed air.
Note that according to Bažant’s theory on the size effect in fracture mechanics, the strength of a rock specimen
decreases with its structural size. Moreover, there is the possibility of a statistical size effect, meaning that the
probability for encountering heterogeneities of low strength increases with increasing sample size (Bažant, 1984).
Such effects couldn’t be identified in this study since only 20 mm diameter samples were tested. All samples
were oriented with their axis perpendicular to bedding. This facilitates the application of the in situ condition
of zero horizontal strain (along the bedding plane) and constant overburden pressure perpendicular to bedding.
Such in situ condition favours the tensile failure along the weak bedding plane. This orientation has also the
particularities that: a) The tensile fracture direction imposed by the experimental conditions coincides with
the bedding plane. b) Radial strains in the direction parallel to bedding are uniform. If the sample is cored
along the bedding plane, it would deform non-uniformly in radial directions in triaxial tests, due to thermo-
poro-elastic anisotropy. In this case, one would need three orientations of strain gages: axial parallel to the
bedding plane, radial parallel to the bedding plane, and radial perpendicular to the bedding plane. Zero radial
strain conditions in this configuration would only be possible, if the radial anisotropy of thermal dilation and
of mechanical contraction is the same. In the next step, the cylinders were cut at desired length (30 - 40 mm)
by using a diamond string saw, to obtain parallel end surfaces. The specimens were afterwards wrapped in an
aluminium foil and covered by a mixture of 70 % paraffin and 30 % vaseline oil, and stored until running the
tests. Small cuttings of the cores were taken after opening the core, to conduct a petrophysical characterisation
(Tab. 2). The volume of the cuttings was measured by hydraulic weighting in petroleum, while the dry density
was obtained after oven drying 24 h at 105 ◦ C. A solid density of 2.69 Mg/m3 , provided by Andra, was adopted
to calculate the porosity φ and the degree of saturation Sr . The specimen suction s was determined by using
a chilled mirror tensiometer (WP4C, Decagon brand). As seen in Tab. 2, the high degrees of saturation (92.5
% and 95.3 %) and low corresponding suction values (24.2 and 17.4 MPa) indicate a good conservation of the
cores and a satisfactory sample quality.
Once removed from storage and unwrapped from aluminium foil and paraffin, the lateral surfaces of the
specimen were protected from drying by wrapping them in an adhesive tape. To ensure correct axial alignment
between the specimen and the end platens, the specimen was placed in a V-block with clamp holders (Fig.Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 11
Table 2 Mean and standard deviation (in brackets) of petrophysical measurements done on cuttings of two COx cores.
ρ ρd φ w Sr s
Core #
[g/cm3 ] [g/cm3 ] [%] [%] [%] [MPa]
EST 2.37 2.22 17.9 7.5 92.5 24.2
1
53650 (0.00) (0.01) (0.2) (0.1) (0.8) (2.1)
EST 2.38 2.21 18.2 7.9 95.3 17.4
2
57185 (0.00) (0.00) (0.2) (0.1) (0.7) (0.1)
3a) and a layer of epoxy resin was applied to both end surfaces. It was observed that the epoxy resin bonded
very well to the claystone, provided that the sample surfaces were previously cleaned from dust by wiping
with a paper towel. Bonding the epoxy resin to the surfaces of the metal platens required more caution: the
surfaces had to be previously roughened with coarse sand paper and wiped with acetone to remove any dirt
or grease. The top and bottom platens were affixed to the specimen with the resin and fastened with clamp
holders. To achieve full bonding at room temperature, the epoxy resin had to cure for about 20 h. The assembly
was then taken from the V-block, the adhesive tape removed and the two strain gages glued to the specimen
at mid-height, along the axial and radial directions (Fig. 2b). A layer of geotextile was wrapped around the
specimen, while ensuring contact between the geotextile and the outlets integrated in the bottom platen. Two
neoprene membranes were superposed over the geotextile, as indicated in Fig. 3b. The strain gages wires were
passed through a section where the membranes overlapped, and sealed with neoprene glue.
a) b)
End-platens Top
Specimen membrane
Strain gage
Specimen
Lead-wire
Neoprene
glue
Bottom
V-block
membrane
Fig. 3 a) Attaching the end platens to the specimen using epoxy adhesive, placed in a V-block to ensure axial alignment; b)
schematic cut to illustrate the layout of two overlapping membranes to pass-through strain gage wires.
In order to prevent any contact of the sample with water before applying the confining stress, to avoid
excessive swelling, the drainage ducts were flushed with air. The specimen was installed in the triaxial cell and
connected to the top and bottom caps, by using the screw connections on the two end platens. O-rings were
put around the membrane to prevent any leaks on both top and bottom caps. Once the strain gage wires were
connected to the data acquisition system, the cell was closed, filled with silicone oil and wrapped by the heating
belt to enable temperature control.
4 Experimental results
In the following sections, the notation of Therzaghi effective stress is used to analyse stress states during
experiments. The effective stress σi0 in a direction i is obtained by σi0 = σi − pf , where σi is the total stress12 Philipp Braun et al.
Table 3 Properties of the specimens measured during the experimental programme, where σ is the isotropic confining stress,
εhyd is the swelling strain during hydration, γt the measured fracture angle with respect to the horizontal bedding plane.
Saturation Initial state Failure
σ εhyd T σz0 0
σh pf T σz0 0
σh pf εz γt
Sample Core [MPa] [%] [◦ C] [MPa] [MPa] [MPa] [◦ C] [MPa] [MPa] [MPa] [%] [◦ ]
EXT1 2 8 0.2 25.0 7.7 6.9 4.9 61.9 -3.6 5.5 19 -0.36 2
EXT2 2 5 0.21 25.0 2.7 3.5 2.1 53.4 -2.4 4.8 7.9 -0.20 18
EXT3 1 12 0.64 35.0 3.2 8.4 4.0 63.5 -3.0 9.6 7.8 -0.33 7
and pf the pore pressure. A seen in Tab. 3, three tests were conducted under different levels of radial effective
stress so as to investigate the effect of radial effective stress on the THM stress-strain behaviour and thermal
failure of the COx claystone. Specimen EXT1 was tested starting from an isotropic effective stress state close
to the in situ one (σz0 = 7.7 MPa, σh0 = 6.9 MPa and pore pressure pf = 4.9 MPa). EXT2 was started at a
lower isotropic effective stress (σz0 = 2.7 MPa, σh0 = 3.5 MPa and pf = 2.1 MPa), while EXT3 was started at
reduced axial effective stress but similar radial effective stress as EXT1 (σz0 = 3.2 MPa, σh0 = 8.4 MPa and pf
= 2.1 MPa).
In tests EXT1 and EXT2, the cell was first brought to a constant initial temperature of 25°C, whereas tests
EXT3 was brought at 35°C. The specimens were submitted to the target isotropic confining stress (5, 8 and 12
MPa, respectively, see Tab. 3) at a loading rate of 0.1 MPa/min. The drainage lines were previously dried and
put under a vacuum of -80 kPa. Once the target confining pressure reached, the drainage lines were saturated
with a synthetic pore water under a pressure of 100 kPa. The composition of this synthetic pore water was
similar to that of the natural COx pore water, according to a composition provided by Andra, which consists
of 1.95 g NaCl, 0.13 g NaHCO, 0.04 g KCl(2H2 O), 0.63 g CaSO4 (7H2 0), 1.02 g MgSO4 (2H2 O), 0.08 g CaCl2
and 0.7 g Na2 SO4 per litre of water. The applied back pressure of 100 kPa was chosen small enough to ensure a
negligible decrease in effective stress and to minimize the poroelastic response of the claystone. Swelling strains
due to hydration stabilized after about two days, with a maximum value εhyd = 0.64% observed for sample
EXT3 under 12 MPa (see Tab. 3). This order of magnitude is comparable to that observed by Belmokhtar
et al. (2017b); Braun et al. (2021b). After hydration, the piston was brought in contact with the specimen and
the desired initial values of the lateral stress (σh0 ), axial stress (σz0 ) and pore water pressure (pf ) were applied
(see Tab. 3). It is assumed that hydraulic equilibrium was achieved when reaching stable deformations.
The thermal extension tests were started by closing the drainage valve V1 (Fig. 2a) and heating the cell
with a constant rate of to 10 ◦ C/h for samples EXT1 and EXT3, and 5 ◦ C/h for sample EXT2. Fig. 4 shows
the relatively constant rates of temperature elevation, while, due to some technical difficulties in the heating
system, heating stopped for a brief time in experiments EXT1 after 4.1 hours and EXT2 after 3.2 hours. In both
cases, the desired rate was recovered afterwards. During heating, the servo-control of the confining pressure
was switched on to maintain radial strains at zero, while the hydraulic pressure in PVC1 was programmed to
keep a constant vertical stress.
Figure 5a presents the data of test EXT1 in terms of applied total stress changes and measured pore pressure
changes, with respect to increased temperature. The resulting measured strains in axial and radial direction
are plotted in Fig. 5b. The data confirms that the servo-control of zero radial strain could be achieved in the
novel device. Note that due to the friction of the sealing gaskets within the axial piston housing, the axial stress
on the specimen did not remain constant, even though the pressure applied by PVC1 (Fig. 2a) was stable.
Nevertheless, these axial stress changes were recorded by the integrated aluminium force transducer (except in
EXT2, where the transducer failed). The direct measurement from the internal force transducer shows some
irregularities and step changes between a minimum axial stress of 10.0 MPa and a maximum value of 15.5 MPa
from the initial value of 12.6 MPa. This results in an uncertainty of approximately ±3 MPa on the axial stress
measured by PVC1. These stress variations are most likely due to the changes in radial stress and temperature,Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 13
70
65
60
55
Temperature [°C]
50
45
40
EXT1
35
EXT2
30
EXT3
25
20
0 1 2 3 4 5 6 7
Time [h]
Fig. 4 Imposed temperature changes with respect to time, applied during the heating tests.
a) b)
30 0.05
EXT1 EXT1
0.00
Total stress and pore pressure [MPa]
25 Total radial stress Radial strain
-0.05
20 -0.10
Extension
-0.15
Strain [%]
15
-0.20
Total axial stress
10 (measured by -0.25
internal force
Axial strain
transducer) -0.30
5 Pore pressure
-0.35
0 -0.40
20 30 40 50 60 70 20 30 40 50 60 70
Temperature [°C] Temperature [°C]
Fig. 5 Test EXT1: a) Total axial stress (measured by force sensor gage and desired to be kept constant by PVC1), total radial
stress (servo-controlled by PVC to constrain zero radial strain) and thermally induced pore pressure. b) Radial strains which
were maintained null by servo-controlling the confining pressure, and axial strains showing a specimen extension.
which affect the movement or static rest of the axial piston, given by the friction of its sealing components.
Unsurprisingly, it is observed that an increase in radial stress is necessary to keep the radial strain to zero
during heating. Increasing radial stress compensates the lateral thermal expansion of the saturated specimen.
Slight irregularities observed in the axial extension strain (Fig. 5b) are a consequence of those observed in the
axial strain control (Fig. 2a). Note that the changes in axial strain are fairly linear until the maximum extension
(–0.36 %) at the temperature at which thermal failure occurred (61.9 ◦ C). A linear increase in pore pressure
from 4.9 MPa to a maximum of 19.0 MPa at 61.9 ◦ C is also observed (Fig. 5a, Tab. 3). Interestingly, all curves
show a loop corresponding to the short temperature drop that occurred at 58 ◦ C.14 Philipp Braun et al.
a) b)
14 0.05
EXT2 EXT2
12 Total radial stress
Total stress and pore pressure [MPa]
0.00
Radial strain
10
-0.05
Extension
8
Strain [%]
-0.10
6
Total axial stress -0.15
4 (from pressure
applied by the
hydraulic system) -0.20 Axial strain
2 Pore pressure
0 -0.25
20 30 40 50 60 70 20 30 40 50 60 70
Temperature [°C] Temperature [°C]
Fig. 6 Test EXT2: a) Total axial stress (measured by PVC1 due to failure of the axial force sensor gage), total radial stress
(servo-controlled by PVC to constrain zero radial strain) and thermally induced pore pressure. b) Radial strains which were
maintained null by servo-controlling the confining pressure, and axial strains showing a specimen extension.
The control of zero radial strains was also ensured in test EXT2 (Fig. 6). In this test, the internal axial force
sensor malfunctioned, therefore only the less precise external axial stress measurement obtained by PVC1 are
shown. Note that this axial stress includes the significant uncertainty of ±3 MPa. The maximum temperature
at thermal failure was 53.4 ◦ C, with a maximum axial extension at εz = –0.20 %. The pore pressure increased
fairly linearly from 2.1 MPa to a maximum of 7.9 MPa (Fig. 6a, Tab. 3). The plateau observed at 40.5
◦
C in the applied temperature (Fig. 4) caused a perturbation in the otherwise linear behaviour. The nonlinear
temperature evolution produced a temporary decrease in total radial stress, axial compaction and pore pressure
stabilization, followed by a recovery of the rates observed before the perturbation.
In experiment EXT3 (Fig. 7), one can observe a decrease in total axial stress from the initial value of 7.2
MPa down to 4.6 MPa at 53 ◦ C, followed by a stabilization with a slight increase above 53 ◦ C, up to 63.5 ◦ C,
at which thermal pressurization failure occurred (Tab. 3). The axial stress variation lies within the uncertainty
of the external control system due to piston friction (±3 MPa). A clear coupling between the axial and the
radial stress is again observed. The increase in pore pressure is almost linear, like in test EXT1. A plateau in
pore pressure changes is observed between 47 and 51 ◦ C, which is supposed to be due to non-linear material
behaviour. Zero radial strain was again achieved, while the axial strains decreased down to εz = -0.34 %.
Figure 8 presents the changes in Terzaghi effective stress with increased temperature. The linear decrease
of the effective axial stress is related to the linear increase in pore pressure observed in Figs. 5 to 7. The
temperatures at which axial effective tensile stress (negative effective stress) starts to be applied are quite
comparable in all three cases, at 43.5 ◦ C for EXT1 and EXT3 and 40.5 ◦ C for EXT2. Finally, as shown in
Tab. 3, thermal tensile failure is observed at -3.6 MPa and 61.9 ◦ C (with a radial effective stress of 5.5 MPa)
for EXT1, at -2.4 MPa and 53.4 ◦ C (with a radial effective stress of 4.8 MPa) for EXT2 and -3.0 MPa at 63.5
◦
C (with a radial effective stress of 9.6 MPa) for EXT3. Interestingly, the changes in effective radial stress are
rather small and limited between the boundary values (5.5 and 8.3 MPa for EXT1, 3.5 and 5.1 MPa for EXT2
and 8.5 and 10.2 MPa for EXT3).
The thermo-hydro-mechanical stress paths of the three tests are represented in Fig. 9 in terms of radial
(y-axis) and axial (x-axis) principal effective stress. The curves indicate that there is no clear dependency of
the thermal tensile failure stress with respect to the applied effective radial stress.Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 15
a) b)
20 0.05
EXT3 EXT3
18
Total radial stress 0.00
Total stress and pore pressure [MPa]
16 Radial strain
-0.05
14
12 -0.10
Extension
Strain [%]
10 -0.15
Pore pressure
8 -0.20
6
-0.25
4 Total axial stress (measured Axial strain
by internal force transducer) -0.30
2
0 -0.35
20 30 40 50 60 70 20 30 40 50 60 70
Temperature [°C] Temperature [°C]
Fig. 7 Test EXT3: a) Total axial stress (measured by force sensor gage and desired to be kept constant by PVC1), total radial
stress (servo-controlled by PVC to constrain zero radial strain) and thermally induced pore pressure. b) Radial strains which
were maintained null by servo-controlling the confining pressure, and axial strains showing a specimen extension.
Note that the aim of this study is to reproduce stress paths similar to the ones in situ, in order to obtain
a failure criterion close to the in situ conditions. Since laboratory and in situ conditions are hardly identical,
undrained conditions are assumed in this study. These are neither exactly met in situ due to the complex 3D
drainage conditions, nor in the laboratory due to the dead volume of the drainage system. In consequence, the
failure criterion measured in this study has to be compared to the results of numerical simulations, which take
into account complex 3D conditions of temperature, stress and drainage.
5 Thermo-poroelastic analysis
Due to the complex nature of the transversely isotropic coupled THM behaviour, it is difficult to directly
evaluate material parameters from the test data. In the presented complex loading paths, a variety of consti-
tutive behaviours (elasticity, plasticity, viscosity, etc.) could be involved. For the following calculations, elastic
behaviour is assumed. This simple analysis shall provide a basis for future in-depth modelling. In the following,
a transversely isotropic THM model is used to simulate the conducted experiments. Material parameters are
taken from different previous studies. The analysis is presented in two steps: First, an element test is simulated
in drained conditions, where the experimentally measured pore pressure is imposed. Only a single material point
is simulated, therefore the system can be solved in an analytical way or using finite differences, if nonlinear
parameters are considered. The simulated stress-strain response is compared with the experimental results, to
calibrate thermo-poroelastic material properties. If one wants to model the material behaviour in undrained
conditions in a second step, the presence of the dead volume of the drainage system has to be taken into
account. This dead volume induced pseudo-undrained conditions, which can be simulated based on previous
calibration tests (Sec. 2.3).16 Philipp Braun et al.
a) b)
12 12
EXT1 EXT2
10 10
8 8
6 6 Radial effective stress
4 Radial effective stress 4
2 2
0 0
Terzaghi effective stress [MPa]
-2 -2
Axial effective stress (measured by
-4 internal force transducer) -4 Axial effective stress (measured by PVC1)
-6 -6
20 30 40 50 60 70 20 30 40 50 60 70
c) Temperature [°C]
12
10 Radial effective stress EXT3
8
6 Axial effective stress (measured by
4 internal force transducer)
2
0
-2
-4
-6
20 30 40 50 60 70
Temperature [°C]
Fig. 8 Changes in Terzaghi effective stresses with temperature during the three heating tests, where negative stresses represent
effective tension. In test EXT2 the internal force sensor malfunctioned and stresses measured by the external transducer are
shown instead.
16
EXT1 EXT2 EXT3
14
Major principal effective stress (radial)
12
10
35.0 °C
63.5 °C
8
[MPa]
25.0 °C
6 61.9 °C
4 53.4 °C
25.0 °C
2
0
-6 -4 -2 0 2 4 6 8
Minor principal effective stress (axial) [MPa]
Fig. 9 Principal (Terzaghi) effective stress paths for all three specimens tested. Circles indicate the initial stress state and
temperature, while crosses mark the specimen failure with their respective temperature.Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 17
5.1 Transversely isotropic thermo-porolastic framework
The thermal properties and THM couplings are analysed within the thermo-poroelastic constitutive equations,
presented among others in the works of Biot and Willis (1957), Palciauskas and Domenico (1982) and Coussy
(2004). The thermo-poromechanical formulations are written for a Representative Elementary Volume (REV)
of the porous material. A relationship between strain εi , total stress σi , pore pressure pf and temperature T is
expressed within this framework, as:
dεi = Cij (dσj − bj dpf ) − αd,i dT (6)
where σj is composed of the normal stresses and shear stresses in different directions:
σi = [σx , σy , σz , σxy , σyz , σzx ]> (7)
The strains in different direction are contained within the strain vector εi :
εi = [εx , εy , εz , εxy , εyz , εzx ]> (8)
In terms of material properties, one has to define the Biot effective stress coefficients bi , the compliance
matrix Cij and the thermal expansion coefficients αd,i .
It has been shown that the COx claystone is transversely isotropic (Plúa et al., 2021b), therefore a simplified
set of coefficients is presented (the properties along the directions x and y are identical, denoted in the following
with a subscript h). Like in most claystones, this plane of isotropy corresponds to the bedding plane. According
to Cheng (1997), the coefficients of transverse isotropy are recalled here.
The vector bi represents the Biot effective stress coefficents in both directions of anisotropy:
bi = [bh , bh , bz , 0, 0, 0]> (9)
One can also write down the entries of the compliance matrix Cij as follows:
C11 C12 C13 0 0 0
C12 C11 C13 0 0 0
C13 C13 C33 0 0 0
Cij = (10)
1/2G0
0 0 0 0 0
0 0 0 0 1/2G 0
0 0 0 0 0 1/2G
with
C11 = 1/Eh
C12 = −νhh /Eh
C13 = −νzh /Ez (11)
C33 = 1/Ez
G0 = Eh /(1 + νhh )
where Ez and νzh are the Young modulus and the Poisson ratio perpendicular to the bedding plane, and Eh
and νhh parallel to the bedding plane, respectively. G describes the independent shear modulus perpendicular
to the isotropic plane.
The thermal behaviour is represented by αd,i , which consists of the two linear drained thermal expansion
coefficients perpendicular (αd,z ) and parallel (αd,h ) to the bedding orientation:
αd,i = [αd,h , αd,h , αd,z , 0, 0, 0]> (12)18 Philipp Braun et al.
The change of fluid content mf is described by Coussy (2004):
dmf 1
= −bi dεi + dpf − (bi αd,i − φαφ + φαf ) dT (13)
ρf M
Here, αφ denotes the bulk thermal expansion coefficient of the pore volume and αf the bulk thermal
expansion coefficient of the pore fluid. The parameter M is the Biot modulus, which, acccording to Aichi and
Tokunaga (2012), can be calculated for a saturated transversely isotropic material:
(1 − νhh ) bh (1 − bz )
1 νzh bz 1 1
= 2(1 − bh ) − + (bz − 2νzh bh ) + φ − (14)
M Eh Ez Ez Kf Kφ
where Kf the bulk modulus of the pore fluid and Kφ the bulk modulus of the pore volume.
5.2 Simulation of a heating test with imposed pore pressure
First, the observed claystone behaviour is simulated during extension using the thermo-poroelastic framework
discussed in Sec. 5.1. Element tests are modelled with homogeneous pore pressure, temperature, stress and
strain distributions. Linear thermo-poroelasticity is considered, therefore the experiments can be reproduced
through analytical calculations. Zero lateral deformation, the measured axial stress, the recorded pore pressure
changes and the applied temperature are imposed. Constitutive parameters were taken from the literature,
summarized in the following: Escoffier (2002) and Belmokhtar et al. (2017b) provided measurements for b close
to 0.9 at effective stresses close to the in situ one (around 9 MPa). Recently, Braun et al. (2021b) estimated the
Biot coefficient also for lower effective stress levels, showing that bz approaches 1.0 at effective stresses tending to
zero. For the following calculations, representing the in situ condition where the mean effective stress decreases,
starting from around 8 MPa, bz = bh = 0.9 was chosen. The Young modulus Ez at 8 MPa mean effective stress
was determined by Menaceur et al. (2015) and Belmokhtar et al. (2018), with values around 3 GPa, and at lower
mean effective stress by Menaceur et al. (2015) and Zhang et al. (2012), with values around 1.0 and 1.5 GPa.
For 8 MPa effective mean stress, Braun et al. (2021b) found through a multivariate regression a best-fit value
of Ez close to 2.6 GPa, which was adopted in this calculation. Less data is available for Eh and the Poisson
ratios. Eh = 5.7 GPa and νzh = 0.11, which was measured by Braun et al. (2021b) under 8 MPa effective
stress is utilized. νhh = 0.29 was given by the Andra database (Guayacán-Carrillo et al., 2017) and confirmed
by Braun et al. (2021b). The anisotropic drained thermal expansion coefficients αd,z = 0.21 × 10−5◦ C−1 and
αd,h = 0.51 × 10−5◦ C−1 are taken from the experimental results of Braun et al. (2021a) on COx claystone.
Moreover, a porosity of 0.18, measured in the present study, is used. It is assumed that all parameters are stress
and temperature independent.
Using Eq. (6), one is able to calculate the changes in radial stress and axial strains, based on the applied
axial stress, the measured pore pressure and the applied temperature (Fig. 10a-f). The radial strains were kept
equal to zero. The strain calculations show a good match with the experimental data, without any parameter
fitting necessary. Only the axial strains in EXT3 become significantly nonlinear at around 50 ◦ C, which is
not represented by the simulation. This difference could be due to natural variability of the specimens, the
occurrence of strain localisation, not detected local strain gages, or induced damage or plastic deformations,
which cannot be captured with the linear thermo-poroelastic model. Calculated radial stresses show a general
slight underestimation. To get a first estimation of the parameter sensitivity, a preliminary parametric analysis
was carried out within the following section.Inducing tensile failure of claystone through thermal pressurization in a novel triaxial device 19
0.05
a) b) c)
0.00 EXT1 EXT2 EXT3
-0.05
-0.10
Axial strain [%]
-0.15
-0.20
-0.25
-0.30
-0.35
-0.40 Experiment
-0.45 Simulation drained
Simulation pseudo-undrained
-0.50
25.0
d) e) f)
23.0
21.0
19.0
Radial stress [MPa]
17.0
15.0
13.0
11.0
9.0
7.0 EXT1 EXT2 EXT3
5.0
21.0
g) h) i)
19.0
17.0
15.0
Pore pressure [MPa]
13.0
11.0
9.0
7.0
5.0
3.0 EXT1 EXT2 EXT3
1.0
20 30 40 50 60 70 20 30 40 50 60 70 20 30 40 50 60 70
Temperature [°C] Temperature [°C] Temperature [°C]
Fig. 10 Experimental results and calculated behaviour using a thermo-poroelastic model on three extension tests EXT1-3. a)
- c) Temperature - axial strain evolution, d) - f) radial stress evolution with temperature and g) - i) thermally induced pore
pressure increase. Note that for the simulation in drained conditions, the pore pressure increase from experimental data was
imposed.20 Philipp Braun et al.
5.3 Simulation of thermal pressurization in pseudo-undrained conditions
To simulate the experiment in pseudo-undrained conditions, the dead volume of the drainage system can
be modelled through Eq. (5), using the drainage system properties evaluated in Sec. 2.3. The change of the
specimen fluid mass per unit volume is calculated through Eq. (13). A single element test with uniform pore
pressure and fluid density within the specimen and the drainage system (pf = pL , ρf = ρL ) is simulated. Next
to the properties used in Sec. 5.2, additional parameters concerning the deformations of the pore space and
the pore fluid are required in Eq. (13). The Biot modulus is computed through Eq. (14). Moreover, as a first
hypothesis, it is assumed that Kφ is equal to the unjacketed bulk modulus of 19.7 GPa (Braun et al., 2021b)
and αφ is equal to 2αh + αz . For the water properties αf and Kf , a dependency on the fluid pressure and
temperature is considered, according to IAPWS-IF97 (2008). In consequence, the material response becomes
nonlinear, which was simulated using the explicit finite difference method with sufficiently small increments
dT ≤ 0.1 ◦ C, dpf ≤ 0.2 MPa.
One observes a good simulation of the axial extension strains in Fig. 10a-c, the radial stress change in Fig.
10d-f and the thermally induced pore pressures in Fig. 10g-i. Axial strain changes are well reproduced by the
simulation, while radial stresses of EXT1 and EXT2 are slightly underestimated. Calculated pore pressures
follow the experimental trend, while they slightly overestimate the result of test EXT1 and underestimate
EXT2. The pore pressure increase of EXT3 is significantly overestimated.
Due to the parallel over- and underestimation of experimental pore pressures, an adjustment of the additional
parameters Kφ , αφ , αf and Kf does not improve the fitting. A preliminary parametric analysis was carried
out by varying separately each model parameter by 10 % and observing the variation of the final pore pressure
in test EXT1. The highest parameter sensitivity was detected for bz , which caused a variation of the resulting
pore pressure of 7 %. The parameters αf and φ caused a variation of 4 % and Ez a variation of 3 %. The
variation of other parameters induced a change of ≤ 1% on the pore pressure result. Also a better fitting of the
horizontal elastic properties Eh and νzh , on which literature data is scarce, could provide better results. For
further information on the in situ variability of parameters, the reader is referred to Plúa et al. (2021b).
The set of linear thermo-poroelastic claystone properties used here appears to be sufficient to simulate
closely the measure laboratory thermal extension. It has to be noted that non-linear water properties are
essential to model thermal pressurization.
6 Failure criterion
Based on the stress state measured at tensile failure (Fig. 9), the existing shear failure criterion of the COx
claystone is extended. A Mohr-Coulomb (MC) type failure criterion, based on uniaxial and triaxial compression
tests on the COx claystone, is given in the Andra database (applied also by Guayacán-Carrillo et al., 2017).
The criterion can be transformed to the principal effective stress domain, written as:
1 + sin ϕ 0 2c cos ϕ
σ10 = σ3 + (15)
1 − sin ϕ 1 − sin ϕ
where σ10 and σ30 are the major and minor principal effective stresses, respectively. Values for the cohesion c =
5.9 MPa and the friction angle ϕ = 23◦ are taken from the Andra database, when the minor effective stress is
perpendicular to the bedding plane. This shear failure criterion, displayed in Fig. 11, provides us an unconfined
compressive strength σc = σ10 (σ30 = 0) = 17.8 MPa. One notes, that the Mohr-Coulomb criterion overestimates
the measured strength in tension, and is therefore not applicable in this region.
Hoek and Brown (1980) and Hoek (1983) described the Hoek-Brown (HB) failure criterion, proposed as an
empirical relationship for observed shear failure of rock under triaxial compressive stress:
0.5
σ 03
σ 01 = σ 03 + σc mi +1 (16)
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